On the Link between Small Ball Probabilities and the Quantization Problem for Gaussian Measures on Banach Spaces

Journal of Theoretical Probability

Volumn 16, Issue 1, 2003, Pages 249-265

  Dereich, S ^a   Fehringer, F ^b   Matoussi, A ^c   Scheutzow, M ^a  

^a TECHNISCHE UNIVERSITÄT BERLIN   (Germany)

^b Université du Maine ^*  (Germany)

^c UNIVERSITÉ DU MAINE   (France)

Author keywords

Gaussian process; High resolution quantization; Isoperimetric inequality; Small ball probability

Indexed keywords

EID: 0037273827     PISSN: 08949840     EISSN: None     Source Type: Journal    
DOI: 10.1023/A:1022242924198     Document Type: Article

References (23)

1. Baldi, P., and Roynette, B. (1992). Some exact equivalents for the Brownian motion in Hölder norm. Probab. Theory Related Fields 93, 457-484.
2. Binia, J. (1974). On the ε-entropy of certain Gaussian processes. IEEE Trans. Inform. Theory 20, 190-196.
3. Bingham, N. H., Goldie, C. M., and Teugels, J. L. (1987). Regular Variation, Cambridge University Press, Cambridge.
4. Bucklew, J. A., and Wise, G. L. (1982). Multidimensional asymptotic quantization theory with r-th power distortion measures. IEEE Trans. Inform. Theory 28, 239-247.
5. Dunker, T. (2000). Estimates for small ball probabilities of the fractional Brownian sheet. J. Theoret. Probab. 2, 357-382.
6. Fehringer, F. (2001). Kodierung von Gaußmaßen, Ph.D. thesis, Technische Universität Berlin.
7. Graf, S., and Luschgy, H. (2000). Foundations of Quantization for Probability Distributions, LNM 1730, Springer, Berlin.
8. Gross, L. (1970). Lectures in Modern Analysis and Applications II, LNM 140, Springer, Berlin.
9. Kuelbs, J. (1976). A strong convergence theorem for Banach space valued random variables. Ann. Probab. 4, 744-771.
10. Kuelbs, J., Li, W. V., and Talagrand, M. (1994). Lim inf results for Gaussian samples and Chung's functional LIL. Ann. Probab. 22, 1879-1903.
11. Kuelbs, J., Li, W. V., and Linde, W. (1994). The Gaussian measure of shifted balls. Probab. Theory Relat. Fields 98, 143-162.
12. Kuelbs, J., and Li, W. V. (1993). Metric entropy and small ball problems for Gaussian measures. J. Funct. Anal. 116, 133-157.
13. Kuelbs, J., and Li, W. V. (1993). Small ball estimates for the Brownian motion and the Brownian sheet. J. Theoret. Probab. 3, 547-577.
14. Ledoux, M. (1996). Isoperimetry and Gaussian Analysis. École d'été de Probabilités de Saint-Flour, LNM 1648, Springer, Berlin, pp. 165-294.
15. p-Norm. Preprint.
16. Li, W. V., and Linde, W. (1999). Approximation, metric entropy and small ball estimates for Gaussian measures. Ann. Probab. 27, 1556-1578.
17. Li, W. V., and Shao Q-M. (2000). Gaussian Processes: Inequalities, Small Ball Probabilities and Applications. Stochastic processes: Theory and methods, handbook of statistics, Vol. 19, Rao, C. R., and Shanbhag, D. (eds.).
18. Li, W. V., and Shao Q-M. (1999). Small ball estimates for Gaussian processes under Sobolev type norms. J. Theoret. Probab. 3, 699-720.
19. Lifshits, M. A. (1995). Gaussian Random Functions, Kluwer, Dordrecht.
20. Luschgy, H., and Pagès, G. (2002). Functional quantization of Gaussian processes. J. Funct. Anal. 196, 486-531.
21. Mason, D. M., and Shi Z. (2001). Small deviations for some multi-parameter Gaussian processes. J. Theoret. Probab. 14, 213-239.
22. Talagrand, M. (1994). The small ball problem for the Brownian sheet. Ann. Probab. 22, 1331-1354.
23. Zador, P. (1982). Asymptotic quantization error of continuous signals and the quantization dimension. IEEE Trans. Inform. Theory 28, 139-149.